SIGNALS AND SYSTEMS I Computer Assignment 2

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1 SIGNALS AND SYSTES I Computer Assignment 2 Lumped linear time invariant discrete and digital systems are often implemented using linear constant coefficient difference equations. In ATLAB, difference equations can be implemented using one of ATLAB's two loop constructs, for loops and while loops, or using ATLAB's built-in filter function. If the system is also a state system, then if can also be implemented using a for loop or a while loop, or using built-in ATLAB functions. Implementing Difference Equations ATLAB's for loop construct is similar to the for loop construct in C and the do loop construct in FORTRAN. ATLAB's for loop construct repeats a set of commands between a for command and its corresponding command a fixed, predetermined number of times. The for loop's syntax is for var = start : increment (default = 1) :, statements For example, for n = -2:2, x(n+3) = n; x generates the signal and for n = 4:-2:-4, y(-n/2+3) = n; y generates the signal x , y The semicolon at the of the statements, x(n+3) = n; and y(-n/2+3) = n;, suppresses the display of the values of x(n+3) and y(-n/2+3), respectively. To implement a difference equation using a for loop, the length of the output must be known a priori. A single output of the difference equation is then calculated for each iteration of the loop. For example, the first 10 output samples of the causal difference equation y(n) a 1 y(n 1) b 0 x(n) b 1 (n 1) x(n) u(n) can be calculated as follows: Computer Homework Chapter 2 Systems. Page 1 of 6

2 x = ones (11,1); y(1) = 0; for n = 2:11, y(n) = -a1*y(n-1) + b0*x(n) +b1*x(n-1); y = y(2:11) If the difference equation is noncausal then the loop is operated in reverse. For example, the first 10 output samples of the noncausal difference equation y(n) a 1 y(n 1) b 0 x(n) b 1 (n 1) x(n) u( n) can be implemented as follows: x = ones (11,1); y(11) = 0; for n = 11:2, y(n-1) = -y(n)/a1 + b0*x(n)/a1 +b1*x(n-1)/a1; y = y(1:10) ATLAB's while loop repeats a set of command between a while command and its corresponding command until a given logical condition is false. The while loop's syntax is while condition, statements For example, n = 0; while n < 5, x(n+1) = n; n = n + 1; x Because of its construct, a while loop can implement a difference equation until a particular event, such as an of data or a button is pressed, occurs. A while loop can also implement a difference equation much like a for loop does. For example, the first 10 output samples of the causal difference equation y(n) a 1 y(n 1) b 0 x(n) b 1 (n 1) x(n) u(n) can be implemented as follows: Computer Homework Chapter 2 Systems. Page 2 of 6

3 x = ones (11,1); y(1) = 0; n = 1; while n <= 11, y(n) = -a1*y(n-1) + b0*x(n) +b1*x(n-1); n = n + 1; y = y(2:11) Difference equations can also be implemented using ATLAB's built-in filter function. The syntax for ATLAB's built-in filter function can be obtained by typing help filter at a ATLAB command prompt. The filter function is part of ATLAB's signal processing toolbox which should be installed on the college systems. If ATLAB reports back that filter.m not found ATLAB's signal processing toolbox has not been installed. Please the system administrator at staff@egr.unlv.edu and inform him of this problem. He might inform you that you need to use the college UNIX systems. Computer Homework Chapter 2 Systems. Page 3 of 6

4 Exercises For Exercises 1-3, use the discrete system described by the difference equation, y(n) a 1 y(n 1) a 2 y(n 2) a 3 y(n 3) b 0 x(n) b 1 (n 1) b 2 (n 2) b 3 (n 3) x(n) is the system's input, y(n) is the system's output and a 0 1 a a a b b b b Draw a) a Direct Form I block diagram of the system. b) a Direct Form II block diagram of the system. c) a transposed Direct Form II block diagram of the system. Indicate the number of delays (memory registers) required to implement each block diagram. 2. Using a for loop or a while loop, write programs that implement each of the block diagrams that you drew in Exercise 1. Indicate the number of delays (memory registers) that each of your difference equation implementations use. Using all three programs, calculate the system's first 51 outputs when the system's input, x(n), is 16 plots on 4 pages, that is, four plots per page.) 3. Using ATLAB's built-in filter function, calculate the system's first 51 outputs when the system's input, x(n), is 8 plots on 2 pages, that is, four plots per page.) Computer Homework Chapter 2 Systems. Page 4 of 6

5 Convolution In general, a lumped linear time invariant system can be described by a linear Nth order constant coefficient difference equation of the form N a k y(n k) b k x(n k) x(n) is the system's input and y(n) is the system's output. If the system is nonrecursive (the output is not a function of the output at other times), then the difference equation can be written as y(n) b k x(n k). This nonrecursive system's impulse response, h(n), can be determined by letting x(n) = (n) which implies that h(n) b k (n k) b n. As a result, difference equations of nonrecursive systems are typically written as y(n) h(k)x(n k) which is the convolution sum for a system with a finite impulse response (FIR) of length +1. Because nonrecursive systems do not have feedback, Direct Form I and Direct Form II implementations of nonrecursive systems are identical. Often, Direct Form implementations of convolution are implemented on array processors or digital signal processors (DSPs). These processor are typically optimized to perform vector matrix operations. As a result, convolution is often expressed as y(n) h(k)x(n k) h T x(n) h T h(0) h(1) h() and x(n) x(n) x(n 1) x(n ) T Computer Homework Chapter 3 Time Domain Analysis of Discrete Linear Systems. Page 2 of 3 onday, August 2, 2004

6 Exercises For Exercises 4-6, use the discrete system described by the finite impulse response, h(n), h(0) h(8) h(2) h(6) h(4) = 0.4 h(1) h(7) h(3) h(5) Draw a) a Direct Form block diagram of the system. b) a transposed Direct Form II block diagram of the system. Indicate the number of delays (memory registers), adds and multiplies required to calculate each output sample. 5. Using the circshift function and vector multiplication, write a program for each of the block diagram that you drew in Exercise 4. Using these programs, calculate the system's first 51 outputs when the system's input, x(n), is 12 plots on 4 pages, that is, three plots per page.) 6. Using ATLAB's built-in conv function, calculate the system's first 51 outputs when the system's input, x(n), is 8 plots on 2 pages, that is, four plots per page.) Computer Homework Chapter 3 Time Domain Analysis of Discrete Linear Systems. Page 3 of 3 onday, August 2, 2004

Lab-Report Digital Signal Processing. FIR-Filters. Model of a fixed and floating point convolver

Lab-Report Digital Signal Processing FIR-Filters Model of a fixed and floating point convolver Name: Dirk Becker Course: BEng 2 Group: A Student No.: 9801351 Date: 30/10/1998 1. Contents 1. CONTENTS...